
In this paper we showed that the primary entanglement has a significant role in decoherence of the system quantum state.

It was shown that if there is no primary entanglement in combined system then the interaction terms which can be responsible for late entanglement are independent. In this case, the interaction term, \(H_{int} \sim \int _{k^{^{\prime }}, k, p} \mathcal {H}^{(2)}_{k k^{^{\prime }} p} \chi _{k} \chi _{k^{^{\prime }}} \phi _{p} \) , contribute to the decoherence.

If we have primary entanglement \(C \ne 0\)  there is a solution of the Schrödinger wave equation, \(N=Q=0\) , the interaction terms are not independent. If we have two of the interaction terms then we can find the others. It is commonly believed that we can add any interaction term between combined system, but this result show that the primary entanglement limit us to choose the interaction term. In this case the decoherence happens like the case of no primary entanglement, \(C = 0\) .

We also find another exact solution \(C,Q \ne 0\) , \(N=0\) , in which the decoherence process is directly affected by entanglement parameter \(C\) . There are two important key points here: first, the last term in the right hand side Eq. (REF ) has no effect on decoherence because the imaginary part of \(F\)  plays role in the decoherence; second, in contrast to the Eq. (REF ) which the imaginary part of \(F\)  is proportional to the integration of \(H_{int} \sim \int _{k^{^{\prime }}, k, p} \mathcal {H}^{(2)}_{k k^{^{\prime }} p} \chi _{k} \chi _{k^{^{\prime }}} \phi _{p} \) , the imaginary part of \(F\)  is related to the integration of \(H_{int} \sim \int _{k^{^{\prime }}, k, p} \mathcal {H}^{(1)}_{k k^{^{\prime }} p} \phi _{k} \phi _{k^{^{\prime }}} \phi _{p} \) . As a result, \( \mathcal {H}^{(1)} \)  has to satisfy the same condition needed for \( \mathcal {H}^{(2)} \)  to have decoherence [1]}, [2]}.

The dependency between the primary entanglement and the interaction terms can have can have a teleological interpretation. Suppose that the semi-classical picture of inflation theory is an effective low energy theory of a universal quantum gravity theory (UQGT). Therefore, both primary entangled states and the interaction terms emerge from the low energy limitation from UQGT. From this prospective, the entanglement and the interaction terms can not be independent.

And at the end we should to emphasis that the difference between theories with different interactions could appear not only their two-points correlation functions at early time but also three point functions are needed. The contributions of these three-points functions come from non-Gaussian part of theories which now is related to entanglement parameter \(C_{k}\)  in the general solution.

